91Ó°ÊÓ

The probability density function (PDF) is crucial in understanding random variables, especially when dealing with continuous distributions, like the exponential distribution. The PDF of a continuous random variable tells us how likely it is for the variable to take on a particular value. For an exponential distribution, which is a common model for the time between events in a Poisson process, the PDF is given by:

• \( f(x) = \lambda e^{-\lambda x} \) for \( x \geq 0 \)

Here, \( \lambda \) is the rate parameter, denoting the average rate at which events occur. In our exercise, both variables \( X \) and \( Y \) follow an exponential distribution with \( \lambda = 1 \), leading to their PDFs as:

• \( f_X(x) = e^{-x} \) for \( x \geq 0 \)
• \( f_Y(y) = e^{-y} \) for \( y \geq 0 \)

These expressions show that the likelihood decreases exponentially as the variable value increases. This property is handy in modeling lifetimes of objects or time until an event happens.

When dealing with two random variables at once, we use the joint probability density function (joint PDF). This function gives the likelihood of both variables simultaneously taking certain values. It's especially straightforward for independent variables, as is the case with \( X \) and \( Y \) in our problem.Because \( X \) and \( Y \) are independent, you can find their joint PDF by simply multiplying their individual PDFs. Mathematically, this is expressed as:

• \( f_{X,Y}(x, y) = f_X(x) \cdot f_Y(y) = e^{-x} \cdot e^{-y} \)

This simplifies to:

• \( f_{X,Y}(x, y) = e^{-(x+y)} \) for \( x \geq 0 \) and \( y \geq 0 \)

The joint PDF \( f_{X,Y}(x, y) = e^{-(x+y)} \) thus illustrates how the combined likelihood of \( X \) and \( Y \) depends exponentially on the sum of both variables. It's particularly useful for understanding and calculating probabilities involving the pairing of \( X \) and \( Y \).

The concept of marginal distribution is pivotal when we want to focus on one variable within a pair of joint variables. It essentially gives us a summary of all potential values for one variable, while "marginalizing" or integrating out the other.In our case, if we're interested in the behavior of \( X \) alone, regardless of \( Y \), we compute the marginal PDF of \( X \). This is done by integrating the joint PDF over all values of \( Y \):

• \( f_X(x) = \int_0^\infty f_{X,Y}(x, y) \, dy \)

Solving gives us back the original exponential distribution for \( X \):

• \( f_X(x) = e^{-x} \) for \( x \geq 0 \)

Similarly, for \( Y \):

• \( f_Y(y) = \int_0^\infty f_{X,Y}(x, y) \, dx \)

This yields:

• \( f_Y(y) = e^{-y} \) for \( y \geq 0 \)

This process shows how, despite considering them jointly initially, both \( X \) and \( Y \) independently retain their distribution characteristics. Understanding marginal distributions is key to simplifying analyses, especially when working with multivariate data.